A Treatise on the Mathematical Theory of Elasticity

Augustus Edward Hough Love

at the University Press, 1906 - 551 ページ

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HISTORICAL INTRODUCTION	1

Scope of History Galileos enquiry Enunciation of Hookes Law Mariottes	7

ANALYSIS OF STRAIN	32

Components of strain	40

ᎪᎡᎢ PAGE 13 Additional methods and results	43

Types of strain a Uniform extension b Simple extension c Shearing strain d Plane strain	44

Relations connecting the dilatation the rotation and the displacement	46

Resolution of any strain into dilatation and shearing strains	47

Inversion	212

Equilibrium of a circular disk under forces in its plane i Two opposed forces at points on the rim ii Any forces applied to the rim iii Heavy disk on h...	213

Examples of transformation	216

THEORY OF THE INTEGRATION OF THE EQUATIONS OF EQUILIBRIUM OF AN ISOTROPIC ELASTIC SOLID BODY 157 Nature of the pro...	217

Resumé of the theory of Potential	218

Description of Bettis method of integration	220

Formula for the dilatation	221

Calculation of the dilatation from surface data	223

Identical relations between components of strain	49

Displacement corresponding with given strain	50

Curvilinear orthogonal coordinates	51

Components of strain referred to curvilinear orthogonal coordinates	53

Dilatation and rotation referred to curvilinear orthogonal coordinates	54

Cylindrical and polar coordinates	56

Introductory	57

Cubical dilatation	59

Reciprocal strain ellipsoid	60

Strain ellipsoid	61

Alteration of direction by the strain	62

Application to cartography	63

Finite homogeneous strain	64

Homogeneous pure strain	65

Analysis of any homogeneous strain into a pure strain and a rotation	67

Simple extension	68

Additional results relating to shear	69

Additional results relating to the composition of strains	70

ANALYSIS OF STRESS 41 Introductory	72

Surface tractions and body forces	73

Equations of motion	74

Equilibrium	75

Measure of stress	77

Transformation of stresscomponents	78

The stress quadric co	79

Additional results	81

The stressequations of motion and of equilibrium	82

Uniform stress and uniformly varying stress	84

Observations concerning the stressequations	85

Graphic representation of stress 58 PAGE 81 81 82 84 85	86

Stressequations referred to curvilinear orthogonal coordinates	87

Special cases of stressequations referred to curvilinear orthogonal coordinates	89

CHAPTER III THE ELASTICITY OF SOLID BODIES Introductory Work and energy	90

Existence of the strainenergyfunction	92

Indirectness of experimental results	94

Hookes	95

Form of the strainenergyfunction	96

Elastic constants	97

Methods of determining the stress in a body 90 90 92 94 95 96 97	98

Thermoelastic equations	106

Initial stress 104 106	109

THE RELATION BETWEEN THE MATHEMATICAL THEORY OF ELASTICITY AND TECHNICAL MECHANICS 76 Limitations of the mathem...	110

Stressstrain diagrams	111

Elastic limits	113

Timeeffects Plasticity	114

Viscosity of solids	115

Eolotropy induced by permanent set 82 Repeated loading	116

Hypotheses concerning the conditions of rupture	117

Scope of the mathematical theory of elasticity	119

ART PAGE 87 Bar bent by couples	126

Discussion of the solution for the bending of a bar by terminal couple	127

SaintVenants principle	129

Equations of equilibrium in terms of displacements	130

Equilibrium under surface tractions only	132

Various methods and results	133

Plane strain and plane stress	134

Bending of narrow rectangular beam by terminal load	136

Equations referred to orthogonal curvilinear coordinates	138

Radial displacement Spherical shell under internal and external pressure Compression of a sphere by its own gravitation	139

Displacement symmetrical about an axis	140

Tube under pressure	141

Application to gun construction	143

Symmetry of structure	146

Geometrical symmetry	147

Elastic symmetry	148

Isotropic solid	152

Classification of crystals	154

Elasticity of crystals	156

110	157

111	158

Extension and bending of a bar	159

113	160

114	161

115	163

116	164

117	166

Uniqueness of solution	167

119	168

Theorem concerning the potential energy of deformation	170

The reciprocal theorem	171

Average strains in an isotropic solid body	172

The general problem of vibrations Uniqueness of solution	173

Flux of energy in vibratory motion	174

Free vibrations of elastic solid bodies	175

General theorems relating to free vibrations	177

Load suddenly applied or suddenly reversed	178

THE TRANSMISSION OF FORCE ART PAGE 129 Introductory	180

First type of simple solutions	182

Typical nuclei of strain	183

Local perturbations	186

Second type of simple solutions	187

Pressure at a point on a plane boundary	188

Distributed pressure	189

Pressure between two bodies in contact Geometrical preliminaries	190

Solution of the problem of the pressure between two bodies in contact	192

Hertzs theory of impact	195

Impact of spheres	197

Effects of nuclei of strain referred to polar coordinates	198

Problems relating to the equilibrium of cones	200

CHAPTER V	201

Displacement corresponding with plane stress	203

Generalized plane stress	205

Force operative at a point	207

Case of a straight boundary	208

Typical nuclei of strain in two dimensions	209

Transformation of plane strain	211

Formulæ for the components of rotation	224

Body bounded by planeFormulæ for the dilatation	225

Body bounded by planeGiven surface displacements	227

Body bounded by planeGiven surface tractions	228

Historical Note	230

Body bounded by planeAdditional results	231

Formulæ for the displacement and strain	232

Recapitulation of the general theory 86 Uniformly varying stress a Bar stretched by its own weight b Cylinder immersed in fluid c Body of any form ...	234

stress in the case of rectangular sections b Extension of this method to curved boundaries c Form of boundary for which the method gives the correct...	235

THE EQUILIBRIUM OF AN ELASTIC SPHERE AND RELATED PROBLEMS 171 Introductory	236

Stress uniform along the beam	237

The sphere with given surface displacements	238

Generalization of the foregoing solution i Integration by means of poly nomials ii Body force required to maintain a state of strain with zero surface ...	239

The sphere with given surface tractions	240

Strain and stress in the elements of the beam	241

Relation between the curvature and the bending moment	242

Conditions restricting the prescribed surface tractions	243

Surface tractions directed normally to the boundary	244

Solution in spherical harmonics of negative degrees	245

Sphere subjected to forces acting through its volume Particular solution	246

Sphere deformed by body force only	247

Gravitating incompressible sphere	248

Deformation of gravitating incompressible sphere by external forces	250

Gravitating nearly spherical body	253

Tidal deformation Tidal effective rigidity of the Earth	255

Plane strain in a circular cylinder	257

Applications of curvilinear coordinates	259

Symmetrical strain in a solid of revolution	260

Symmetrical strain in a cylinder	263

VIBRATIONS OF SPHERES AND CYLINDERS 190 Introductory	265

Solution by means of spherical harmonics	266

Formation of the boundary conditions for a vibrating sphere	268

Incompressible material	271

Vibrations of the first class	272

Vibrations of the second class	273

Further investigations of the vibrations of spheres	274

Vibrations of a circular cylinder	275

Torsional vibrations	276

Transverse vibrations	278

THE PROPAGATION OF WAVES IN ELASTIC SOLID MEDIA ART PAGE 203 Introductory	281

Motion of a surface of discontinuity Kinematical conditions	283

Motion of a surface of discontinuity Dynamical conditions	284

Velocity of waves in isotropic medium	285

Velocity of waves in æolotropic solid medium	286

Wavesurfaces	287

Motion determined by the characteristic equation	289

Arbitrary initial conditions	291

Motion due to body forces	292

Additional results relating to motion due to body forces	294

Waves propagated over the surface of an isotropic elastic solid body	295

TORSION 215 Stress and Strain in a twisted prism	298

The torsion problem	299

Method of solution of the torsion problem	301

Analogies with Hydrodynamics	302

Distribution of the shearing stress	304

Solution of the torsion problem for certain boundaries	305

Additional results	306

Graphic expression of the results	308

Analogy to the form of a stretched membrane loaded uniformly	310

Torsion of solotropic prism	312

THE BENDING OF A BEAM BY TERMINAL TRANSVERSE LOAD 227 Stress in a bent beam	314

Statement of the problem	315

Necessary type of shearing stress	316

Formula for the displacement	318

a The circle b Concentric circles c The ellipse d Confocal ellipses e The rectangle f Additional results	322

a Curvature of the strained centralline	323

b Neutral plane c Obliquity of the strained crosssections d Deflexion e Twist ƒ Anticlastic curvature g Distortion of the crosssections into curved surfa...	326

Distribution of shearing stress	327

a Asymmetric loading b Combined strain c Eolotropic material	330

334	336

THE THEORY OF CONTINUOUS BEAMS 347 245 Extension of the theory of the bending of beams	350

The problem of continuous beams 350	353

Single span a Terminal forces and couples b Uniform load Supported ends c Uniform load Builtin ends d Concentrated load Supported ends e Conce...	357

Graphic method of solution of the problem of continuous beams	360

Development of the graphic method	362

CHAPTER XVIII	365

Kinematical formulæ	366

Equations of equilibrium	371

The ordinary approximate theory	372

Nature of the strain in a bent and twisted	373

Approximate formulæ for the strain	376

Discussion of the ordinary approximate theory	377

Rods naturally curved 365	379

Stability of inflexional elastica	393

Rod bent by forces applied along its length	402

Introductory	408

Rod free at one end and struck longitudinally at the other	415

The whirling of shafts	421

Problems of equilibrium a Incomplete circular ring bent in its plane	429

Relation between the flexural couples and the curvature	442

Plate bent to a state of plane stress	449

Plate bent by pressure uniform over a face	456

Plate bent by uniformly varying pressure and clamped at the edge	463

INEXTENSIONAL DEFORMATION OF CURVED	471

366	478

Inextensional vibrations i Cylindrical shell ii Spherical shell	485

Extension and curvature of the middle surface of a plate or shell	491

ᎪᎡᎢ PAGE	498

Second approximation in the case of a curved plate or shell	506

Vibrations of a thin cylindrical shell a General equations b Extensional	516

Vibrations of a thin spherical shell	522

Problems of stability a Buckling of a rectangular plate under thrusts in	529

Applications of the method of moving axes	536

INDEX	543

370	548

379	549

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A Treatise on the Mathematical Theory of Elasticity, 第 2 巻
Augustus Edward Hough Love
全文表示 - 1893

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多く使われている語句

applied Article axis body forces central-line centre coefficients components of strain constants coordinates corresponding cos² couple cross-section curvature curve cylinder deflexion denote determined dilatation direction displacement dx dy elastic equations of equilibrium expressed formulæ given harmonic function integral isotropic load Lord Rayleigh Math middle surface motion normal obtained P₁ parallel Phil plane strain plane stress plate Poisson's ratio pressure principal axes problem quadratic function quadric quantities R₁ radius right angles rotation satisfy the equations shearing shearing stress sin² solid solid harmonics solution sphere spherical strain-components strain-energy-function stress stress-components stress-system surface tractions symmetry tangent tension theorem theory torsion unstrained values vanish velocity vibrations Young's modulus Z₂ θα λ+2μ λ+μ дах дв дг др дф дх дх ду

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書誌情報

書籍名	A Treatise on the Mathematical Theory of Elasticity A Treatise on the Mathematical Theory of Elasticity, Augustus Edward Hough Love
著者	Augustus Edward Hough Love
版	2
出版社	at the University Press, 1906
書籍の提供元	ニューヨーク公共図書館
デジタル化された日	2006年4月28日
ページ数	551 ページ

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